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Cohomology of Fuchsian groups and Fourier interpolation

Mathilde Gerbelli-Gauthier, Akshay Venkatesh

TL;DR

The paper constructs an abstract, cohomological framework linking Fourier interpolation to the first cohomology of a Fuchsian group with coefficients in the Weil oscillator representation. It provides two complete proofs of the key main theorem: (i) a Bunke–Olbrich–style principal-series resolution with Laplacian surjectivity, and (ii) a direct $(\mathfrak{g},K)$-module extension approach, both reducing interpolation to cohomological multiplicities and Ext computations. The work clarifies how the interpolation data at the lattice points arises from cohomology, yields an exact surjectivity result for the Casimir operator on automorphic forms, and extends the method to variants including odd Schwartz functions, higher dimensions, and Heisenberg-uniqueness phenomena. The results connect explicit interpolation formulas to deep representation-theoretic structures, offering a unifying perspective with potential extensions to nonlattice groups and broader coefficient systems.

Abstract

We give a new proof of a Fourier interpolation result first proved by Radchenko-Viazovska, deriving it from a vanishing result of the first cohomology of a Fuchsian group with coefficients in the Weil representation.

Cohomology of Fuchsian groups and Fourier interpolation

TL;DR

The paper constructs an abstract, cohomological framework linking Fourier interpolation to the first cohomology of a Fuchsian group with coefficients in the Weil oscillator representation. It provides two complete proofs of the key main theorem: (i) a Bunke–Olbrich–style principal-series resolution with Laplacian surjectivity, and (ii) a direct -module extension approach, both reducing interpolation to cohomological multiplicities and Ext computations. The work clarifies how the interpolation data at the lattice points arises from cohomology, yields an exact surjectivity result for the Casimir operator on automorphic forms, and extends the method to variants including odd Schwartz functions, higher dimensions, and Heisenberg-uniqueness phenomena. The results connect explicit interpolation formulas to deep representation-theoretic structures, offering a unifying perspective with potential extensions to nonlattice groups and broader coefficient systems.

Abstract

We give a new proof of a Fourier interpolation result first proved by Radchenko-Viazovska, deriving it from a vanishing result of the first cohomology of a Fuchsian group with coefficients in the Weil representation.
Paper Structure (33 sections, 14 theorems, 98 equations)

This paper contains 33 sections, 14 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Theorem \ref{['mainthm']} implies Theorem \ref{['interpolation']}
  3. The proof of Theorem \ref{['mainthm']}.
  4. Questions
  5. Acknowledgements
  6. Covering groups of $\mathrm{SL}_2(\mathbf{R})$
  7. Lie algebra
  8. Iwasawa decomposition
  9. $(\mathfrak{g}, K)$-modules
  10. Globalizations
  11. First proof of Theorem \ref{['mainthm']}: resolutions of principal series.
  12. Setup
  13. Second proof of Theorem \ref{['mainthm']}: extensions of $(\mathfrak{g}, K)$-modules
  14. Proof of Theorem \ref{['mainthm']}: reduction to $(\mathfrak{g}, K)$ extensions
  15. Evaluation of the $(\mathfrak{g}, K)$-ext
  16. ...and 18 more sections

Key Result

Theorem 1.1

The map is an isomorphism onto the codimension 1 subspace of $\mathfrak{s} \oplus \mathfrak{s}$ cut out by the Poisson summation formula, i.e. the subspace of $(x_n, y_n)$ defined by $\sum_{n \in \mathbf{Z}} x_{n^2}= \sum_{n \in \mathbf{Z}} y_{n^2}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 13 more